Regarding just the sweep oscillators, ignoring effects on the video signal itself:
Depending on how far back you go -- the standard method AFAIK is a blocking oscillator. Roughly speaking, this is a type of relaxation oscillator, which stays latched (on/off) until its timing element passes a threshold, then it toggles state.
If we superimpose ("inject") a signal on top of that timing signal (or the threshold, equivalently), we can make it trigger sooner, and in sync with the source.
More concretely: suppose there is an oscillator, with a voltage ramp signal of say 0 to 5V. If we superimpose a 0.5V sync signal on it, then for injected frequencies within 0.5/5 = 10% of normal (actually, -10 to 0%, since the injected signal can only make it fire sooner than usual), it will lock 1:1. Or within 20% of 2x, it will lock 2:1, etc. Note that it won't lock 9:1 vs. 10:1, at least not consistently: the difference in threshold between the 9th and 10th pulses will not be well defined, and it'll probably lock to the earlier one instead.
(This is all very hand-waved, mind. Driven oscillators are, in general, chaotic systems, so technically just about anything is possible, given just the right circuit. The usual circuits, adjusted properly, I don't think are too pathological, so this should do for basics.)
Exactly what thresholds and ratios are used, likely vary between TV sets. Also likely they're defined by component tolerances, so, 5 and 10% resistors and capacitors being common back in the day, they probably chose a similar injection level. That is, NTSC itself (or whatever else) might be very precisely defined, but the time constants in the set cannot be made so tight (with economical manufacture), so they set it on the loose side so it needs less adjustment. (The HOLD controls adjust the oscillator's natural frequency, so this will even be trimmed by the user.)
I don't know that other methods were used much if at all, e.g. a fully fleshed-out PLL system? Or like a one-shot timer (ramp starts after sync pulse), which might technically do given a valid video signal, but since all the tube voltages were derived from the horizontal sweep system, at least horizontal had to keep running (and for that matter, vertical had to keep running to prevent burn-in of a line on the tube, so the same goes for that).
Digital sets (90s+?) likely measure the timing directly, and whether they accept or reject pulses between expected intervals, I don't know. Of course a multisync (computer) monitor might well try and lock to it (1:1); or just complain that it's an illegal mode.
So, a 3:1 ratio probably is fine with most analog sets. It's really weird putting in extra sync pulses though ("blacker than black" level during a sweep?).